6942
J. Phys. Chem. B 2002, 106, 6942-6946
Increasing Cluster Correlations during Electrochemical Insertion Unfolded by the Correlation Correction Factor in the Frame of the Cluster Variation Method C. Pe´ rez Vicente* and J. L. Tirado Laboratorio de Quı´mica Inorga´ nica, UniVersidad de Co´ rdoba, Campus de Rabanales, Edificio C3, 14071 Co´ rdoba, Spain ReceiVed: NoVember 9, 2001; In Final Form: April 23, 2002
Insertion compounds are the most promising materials for advanced rechargeable batteries. The electrochemistry of the insertion process is described here by using a novel application of the cluster variation method (CVM). This approach allows us to firmly establish the role of correlation in the analysis of the multiple plateau found in cell potential vs composition curves obtained for different host solids. The model is successfully applied to lithium cells using layered SnS2 or spinel-related LiMn2O4 solids as the active material of the positive electrode.
1. Introduction A precise knowledge of the thermodynamics of a reaction in which a solid lattice with ion-transport properties is involved, such as insertion reactions, is of prime interest with a view to its practical application as active electrode material in advanced energy storage systems. The large number of compounds studied for these purposes increases continuously. Dichalcogenides, such as TiS2,1,2 oxides with a LiMO2 stoichiometry3,4 and the spinelrelated LiMn2O45,6 have proven of particular interest. Particularly LiCoO2 and carbon are relevant examples by the fact of being the active electrode materials of the first commercial battery product of “Li-ion” technology.7 The Helmholtz free energy (A) of an insertion system can be expressed as
A ) E - TS ) sNB + rNBB - kT ln W(N,NB,NBB) (1) where s and r are the energy of stabilization of the occupied site and the repulsion between coupled inserted ions, respectively; NBB is the number of coupled atoms, k is the Boltzman constant, T is the temperature, and W is the number of different configurations to arrange NB atoms in N sites. Several models have been developed to approximate W. The lattice-gas model is the most common model used to describe intercalation compounds.8-10 Another interesting approach based on the Fermi-Dirac statistics was developed by allowing the term of interactions between inserted ions, based on the model of Fowler and Guggenheim.11 When permanent electronic conduction is present and the extension of the insertion process is limited by the ionic occupancy, it is equivalent to the lattice-gas model.12 On the other hand, although the Ising model was originally used as for ferromagnetism, it can also be used to describe other systems, such as binary alloys and lattice-gas systems. The Bethe approximation was applied to the Ising model to interpret the behavior in systems with cooperative interactions, such as ferromagnetic spin lattices. The entropy equivalent to this approximation was given by Kikuchi.13 More recently, Kudo and Hibino5 derived chemical potentials by assuming different forms of configurational entropy in intercalation systems with * Corresponding author. E-mail:
[email protected]. Fax: + (34) 957 218 621.
Figure 1. Triangular lattice showing equivalent sets of R, β, and δ sites.
repulsive interactions applied to LixMn2O4. They found that the Bethe entropy fits well to the unknown exact entropy when inserted cations are close to a random distribution. But it fails in explaining the boundary regions, when B-B interactions are forced or avoided, due to strong cooperative interactions. A theory of cooperative phenomena was developed by Kikuchi,13 known as cluster variation method (CVM), which takes into account the symmetry of the lattice. The concept of correlation correction factor (CCF) allows a simple and systematic formulation of CVM. This approach is used to study the insertion in some compounds used as cathode material in secondary batteries. 2. Intercalation in a Triangular Lattice A. Ordered Phase at 1/2. Consider a triangular lattice (Figure 1) with N equivalent points. Assuming that R, β, and δ points are equivalent, the CVM allows the evaluation of the number of different configurations in the lattice, defined as Wlattice, according to the expression14
Wlattice ) (Wpoint)N(Gpair)3N(Gtriangle)2N )
{pair}3N
{point}N{triangle}2N
(2)
where Wpoint is the number of the different possible configurations of arranging B object in N sites, Gpair takes into account
10.1021/jp0141171 CCC: $22.00 © 2002 American Chemical Society Published on Web 06/06/2002
Cluster Correlations during Electrochemical Insertion
J. Phys. Chem. B, Vol. 106, No. 27, 2002 6943
TABLE 1: Distribution of B-V Pairs, Their Probabilities, and Multiplicities Corresponding to the {Pair} Term distribution
probability
multiplicity (γ)
B-B B-V V-V
y1 y2 y3
1 2 1
TABLE 2: Distribution of Triangular Configurations, Their Probabilities, and Multiplicities, Corresponding to the {Triangle} Terma
Figure 2. Plot of -r/kT as a function of the intercalation ratio for the order-disorder transition in a triangular lattice.
a
Solid and open circles represent B and V, respectively.
the possible correlation in the formation of pairs object-object (B-B), object-vacancy (B-V), and vacancy-vacancy (VV), and Gtriangle considers the possible correlation in the formation of triangles B-B-B, B-B-V, B-V-V, and V-V-V. Thus, Gpair and Gtriangle evaluate the possible formation of clusters of two and three objects. The terms {point}, {pair}, and {triangle} are defined as
{point} ) Π(xiN)
(3a)
{pair} ) Π(γiyiN)!
(3b)
{triangle} ) Π(ωiwiN)!
(3c)
where x1 is the fraction of occupied sites x1 ) B/N and x2 is the fraction of unoccupied sites x2 ) (N-B)/N, γ and y represent the multiplicity and the probability of the different {pair} configuration, as defined in Table 1, and ω and w represent the multiplicity and the probability of the different {triangle} configuration, as defined in Table 2. From the Helmholtz free energy calculated according to eq 1, the cell voltage is obtained as V ) -(1/F)‚(∂A/∂x1), where F is the Faraday’s constant. The Helmholtz free energy is minimized according to ∂A/∂pi ) 0, where pi parameters are used to make a systematic parametrization, according to the Sanchez and Fontaine’s rules.15 It is based on the use of the operator Γi(q) ) 1/2[1 + iσ(q)], where “i” takes values 1 and -1 for each one of the two possible events (object or vacancy). σ(q) is the occupancy operator at lattice point q that takes the values σ(q) ) 1 if i ) 1, and σ(q) ) -1 if i ) -1. Thus, Γi(q) ) 1 if the event “i” takes place at lattice point q and zero otherwise. The concentration of any cluster in a given configuration can be written as
ξij...k )
1 Mij...k
∑Γi(q1) Γj(q2) ... Γk(qn)
(4)
where ξij...k refers to xi, yi, and wi, Mij...k is the total number of clusters in the lattice, and the summation is performed over all lattice points. Equation 4 can be rewritten as
ξij...k )
1 (1 + 2n
∑ fij...k pij...k)
(5)
where pij...k are the n-cluster correlation parameters, and fij...k is a product involving the ij...k products arising from the product of Γi(q) operators. From these relationships, pij...k can be obtained. In the case of the regular triangular lattice, with {point}, {pair}, and {triangle} clusters, it gives
p1 ) x1 - x2
(6a)
p2 ) y1 - 2y2 + y3
(6b)
p3 ) w1 - 3w2 + 3w3 - w4
(6c)
In electrochemical reactions, the partial derivative ∂A/∂p1 is proportional to the voltage of the cell. The Helmholtz free energy is then minimized according to ∂A/∂pi ) 0 (i ) 2, 3), giving the following restrains:
( )( ) ( )
z1z4 ∂A )0w ∂p2 z2z3
3
r y22 3 ) exp y1y3 RT
z1 z33 ∂A )0w )1 ∂p3 z z3
(7a)
(7b)
4 2
According to Sanchez and Fontaine,15 the critical temperature is determined by the equation15
| |
∂2 A )0 ∂pi∂pj
(8)
where i, j ) 2, 3. Combining eqs 7 and 8, the condition for the order-disorder transition is obtained. Figure 2 plots -r/kT as a function of the intercalation ratio. Such ordering has been found in LiCoO2.16 At x1 ) 0.5, and T ) 333 K, the voltage of the order-disorder transition was determined to be ca. 4.12 V. It allows us to calculate s ) -389 kJ/mol and r ) -8.49 kJ/mol. B. Ordered Phases at 1/3 and 2/3. Many layered compounds used as insertion materials possess a layered structure, where the interlayer space contains a triangular lattice of octahedral sites available for insertion. During lithium intercalation/ deintercalation some ordered phases usually appear. As a consequence of B-B pair avoidance, a frequent arrangement is obtained in which the intercalated amount is one-third of the total available sites. When eq 2 is used to calculate the entropy, it gives negative values at x1 ) 1/3 (see Table 3). Thus, even if the classical CVM of a triangular lattice gives good results to
6944 J. Phys. Chem. B, Vol. 106, No. 27, 2002
Vicente and Tirado
TABLE 3: Values of S/RT at Some Selected Values of Intercalation Ratio, Calculated According to Eq 2 and Eq 9a imposing the condition of B-B avoidance intercalation according to ratio eq 2 0 /6 0.2961 1 /3 1
0 0.3328 0 -0.4621
according to eq 9a φ ) 0 φ ) 0.5 φ ) 1 0 0.1155 0.0582 0
0 0.2747 0.2674 0.2027
random distribution
0 0.4338 0.4766 0.4055
0 0.4506 0.6075 0.6365
analyze random distribution and order-disorder transitions at x1 ) 1/2, it fails to explain other ordered phases. As shown in Figure 1, the sites denoted R form a triangular sublattice. Similar sublattices are formed by sites β and δ. On the assumption that the three sublattices are not equivalent, eq 2 applies for each triangular sublattice. If we add a term to take into account the CCF between them, i.e., the ij pairs (Rβ, βδ, and δR) and the Rβδ triangles formed, the expression is (R (β (δ φ (WRβδ)6 ) Wlattice ‚Wlattice ‚Wlattice ‚(W(Rβδ corr )
(9a)
W(Rβδ corr ) {pair - Rβ}3N{pair - βδ}3N{pair - δR}3N {point - R}N{point - β}N{point - δ}N{triangle - Rβδ}6N (9b) where the term W(Rβδ corr is the CCF between sublattices and φ takes into account the statistical weight of the CCF factor (0 e φ e 1). The terms {pair - ij} and {triangle - Rβδ} were (i (j calculated according to the following rules: y(ij 1 ) x1 x1 and (Rβδ (R (β (δ z1 ) x1 x1 x1 . The intercalation ratio is the defined as x1 ) (β (δ (x(R 1 + x1 + x1 )/3. The above restrictions are equivalent to a random distribution in each sublattice, while the nonrandom distribution in the complete lattice is introduced by a different filling of each sublattice. This approach gives nonnegative entropy for B-B avoidance when x1 f 1/3, in opposition to the classical expression in eq 2. When the three sublattices are considered as equivalent and φ ) 1, then (R (β (δ ) Wlattice ) Wlattice ) Wlattice Wlattice
(10a)
{point - R} ) {point - β} ) {point - δ} ) {point} (10b) {pair - Rβ} ) {pair - βδ} ) {pair - δR} ) {pair} (10c) {triangle - Rβδ} ) {triangle} W(Rβδ corr )
{pair}9N {point}3N{triangle}6N (WRβδ)6 ) (Wlattice)6
) Wlattice3
(10d)
(10e)
(10f)
Thus, the left-term exponent 6 in eq 9a is introduced to normalize the final expression, and when a random distribution in assumed, and φ ) 1, eq 9a gives the more simple relationship expressed in eq 2. Figure 3a shows three plots of voltage vs intercalation ratio obtained when B-B interactions are avoided for x1 e 1/3 and then minimized, corresponding to different
Figure 3. (a) Plots of voltage vs intercalation ratio obtained with no interactions for x1 e 1/3 and minimized interactions for x1 > 1/3, and three values of correlation between sublattices: φ ) 0, 0.5, and 1. (b) Plots of voltage vs intercalation ratio obtained at φ ) 1 for three cases: (i) as in (a); (ii) no interactions for x1 e 1/3, and a random distribution for x1 > 1/3 (ordered phases only at x1 ) 1/3); (iii) a random distribution in the three sublattices (no ordered phase).
values of correlation between sublattices: φ ) 0, 0.5, and 1. The voltage was calculated as
V)-
1 ∂A F ∂x1
A ) E - TS
(11a)
s r (Rβ (γ (βδ (δR E ) (x(R + x(β 1 + x1 ) + (y1 + y1 + y1 ) 3 1 3 S ) R ln WRβγ (11b) The energetic terms were calculated with s/kT ) -100 and r/kT ) + 10. The profile is the same for all cases, although the voltage steps are less steep when φ increases. The increase in φ does not change the mean values of the voltage either of the central plateau or of the critical points at x1 ) 1/3, 2/3. On the contrary, the voltages of the first and third plateau are slightly increased and decreased, respectively. Figure 3b shows three plots of voltage vs intercalation ratio obtained at φ ) 1 for three cases: (i) as in Figure 3a, where the three plateaus are present; (ii) when interactions are avoided for x1 e 1/3 and then allowed to a random distribution (ordered phases only at x1 ) 1/3), resulting in a first plateau extending to x1 ) 1/3, where the complete ordered phase appears, and a second extending to x1 ) 1, where the interlayer spacing is completely filled; (iii) where there is a random distribution in the three sublattices, where no ordered phase appears, resulting in a continuous decrease of voltage, without any plateau, in the 0 e x1 e 1 interval. Figure 4 shows the experimental17 and simulated curves when SnS2 is used as cathode material vs Li. If no correlation is assumed (φ ) 0) the first plateau fits well, but not the second.
Cluster Correlations during Electrochemical Insertion
J. Phys. Chem. B, Vol. 106, No. 27, 2002 6945
Figure 5. (a) Plots of entropy vs intercalation ratio for four cases: when filling is random (φ ) 1) and when interactions are avoided (φ ) 0, 0.5, and 1). (b) Plots of cell voltage vs intercalation ratio.
Figure 4. Experimental (from ref 17) and simulated curves of cell voltage vs intercalation ratio when SnS2 is used as cathode material vs Li. The simulated curves were obtained from eq 9a, with φ ) 0 (A), φ ) 1 (B), and φ ) x1 (C).
It is indicative of small or null correlation at the beginning of the insertion. The opposite effect is observed when complete correlation is imposed (φ ) 1), where the second plateau is fairly well fitted but not the first, now indicating that strong correlation exists and the end of the insertion. Then, we assumed an increasing correlation between sublattices (φ ) x1). The simulated and experimental curves agree well in both the first and second plateaus. It shows that Li insertion results in an increasing correlation between sublattices. 3. Intercalation in Spinel Host Structures Transition metal oxides with spinel structure are also widely used as cathode material in secondary batteries. They have typical stoichiometries LiM2O4, where Li occupies one-eighth of the tetrahedral sites. The lattice of tetrahedral sites can be interpreted in terms of two interpenetrated sublattices, each one corresponding to a fcc structure. We can consider that the two sublattices are not equivalent, and the entropy of the lattice is expressed as a combinations of the entropy of each sublattice and a correlation factor between them: (β (Rβ φ (WRβ)4 ) W(R lattice‚Wlattice‚(Wcorr)
W(Rβ corr )
{point - R}{point - β} {pair - Rβ}2
(12a) (12b)
where the term W(Rβ corr is calculated by taking into account the close interactions between sites of both sublattices. The (β intercalation ratio is defined as x1 ) (x(R 1 + x1 )/2, with 0 e (i x1 e 1, while terms {pair - Rβ} were calculated according to (R (β y(Rβ 1 ) x1 x1 . The entropy is calculated as S ) R ln WRβ. Figure 5a shows the changes in entropy as a function of the intercalation ratio for four cases: when there is a random filling (φ ) 1) and when B-B interactions are minimized, for φ ) 0, 0.5, and 1. In the last three cases the entropy shows a minimum at x1 ) 0, corresponding to the ordered phases where one of the sublattices is filled but not the other. The higher correlation values of φ result in higher entropy values. Concerning the cell voltage, it has been calculated as
V)E)
1 ∂A F ∂x1
s (R (Rβ (x + x(β 1 ) + ry1 2 1
A ) E - TS S ) R ln WRβ
(13)
and is shown in Figure 5b. The increase in φ does not change the mean values of the voltage either of the two plateaus or of the critical point at x1 ) 0.5. On the contrary, the slope of the curve over the plateaus is affected, slowly increasing with φ. Figure 6 shows the experimental18 and simulated curves when LiMn2O4 is used as cathode material vs lithium. As in the case of SnS2, if no correlation is assumed (φ ) 0), the first plateau of the discharge curve is well fitted, but not the second, indicating that low correlation is present for low lithium content. The opposite is observed for complete correlation (φ ) 1), where the second plateau is fairly well fitted but not the first. It allows us to estimate that strong correlation exists for high Li content. When an increasing correlation is assumed (φ ) y1), the simulated curve agrees fairly well with both plateaus, showing that the correlation between sublattices increases with Li content.
6946 J. Phys. Chem. B, Vol. 106, No. 27, 2002
Vicente and Tirado variation method (CVM). This approach established the role of correlation in the analysis of the multiple plateaus found in cell potential vs composition curves obtained for different host solids and is successfully applied to describe the lithium insertion in layered SnS2 and spinel-related LiMn2O4 solids, used as an active material of the positive electrode in lithium cells. This approach allows non-negative values of the entropy for the extreme case of avoidance of B-B interactions, in opposition to the classical description based on CVM. Its application to the above cited cases (SnS2 and LiMn2O4) shows that correlations to form a pair or triangle cluster are negligible at the beginning of the insertion, when the lithium content is low. On the contrary, further lithium insertion results in short-range interaction between lithium atoms, with the formation of pair and triangle clusters, as revealed by the increase of the statistical weight of the correlation correction factor. Acknowledgment. This work was supported by MCYT (MAT1999-0741 and MAT2000-2721-CE). References and Notes
Figure 6. Experimental18 and simulated curves of cell voltage vs intercalation ratio when LiMn2O4 is used as cathode material vs Li. The simulated curves were obtained from eq 12a, with φ ) 0 (A), φ ) 1 (B), and φ ) y1 (C).
In conclusion, the electrochemical insertion process is described by using a novel approach in the frame of the cluster
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